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Formats of solutions

A solution list 
of a complex polynomial system is denoted by the number of solutions and the dimension, followed by, for each solution, its position in the list, and the solution itself. The solutions are separated by a banner line.
A solution 
consists of the current value of the continuation parameter t, its multiplicity (or winding number) m, and the solution vector.

A solution vector 
contains as many lines as the dimension. The i-th line starts with the symbol that represents the i-th unknown, followed by the colon `:' and two floating-point numbers representing respectively the real and imaginary part of the solution component.

As example we list the solution list of the regular solution (1,i) of a 2-dimensional system in the unknowns x and y at t=1.

1 2
solution 1 :
t :  1.00000000000000E+00  0.00000000000000E+00
m : 1
the solution for t :
 x : 1.00000000000000E+00  0.00000000000000E+00
 y : 0.00000000000000E+00  1.00000000000000E+00

The separation banner between two solutions may contain additional information on output. The following is an example of a last line of a list of solutions as produced by the path-following routines:

== #regu : 4 = #sing : 0 = #clus : 0 = #infi : 0 = #fail : 0 ==
This means that four paths have yielded four regular solutions. There were no singular and no clustered solutions, no solutions at infinity, nor path failures.

Newton's method used in the root refiner produces separation banners like

== err :  9.930E-17 = rco :  1.998E-01 = res :  4.441E-16 = real regular ==
to indicate that the norm of the last correction vector of Newton method is 9.930 10-17, that the inverse of the condition number   of the Jacobian matrix equals 1.998 10-1, and that the residual equals 4.441 10-16. By its condition number, the solution is regular and, since the imaginary parts of the solution components are sufficiently small, the solution is reported to be real. Separation banners have no meaning to the program when the solution lists are submitted as input.

When a start system is submitted to the program, the list of solutions is expected to be after a banner that contains the word SOLUTIONS.

next up previous index
Next: Index Up: Input/output formats Previous: Formats of polynomial systems
Jan Verschelde